Pigeons, specifically the fantail pigeon and the pouter pigeon, are a motif in the book. For example, in Chapter 10, Darwin uses these pigeons to make the point that "intermediate forms" between two related species can often be more readily found among ancestors than among living species:
To give a simple illustration: the fantail and pouter pigeons are both descended from the rock-pigeon; if we possessed all the intermediate varieties which have ever existed, we should have an extremely close series between both and the rock-pigeon; but we should have no varieties directly intermediate between the fantail and pouter; none, for instance, combining a tail somewhat expanded with a crop somewhat enlarged, the characteristic features of these two breeds.
Darwin is able to trace both the fantail and the pouter to a common ancestor, the rock-pigeon. (Modern genome sequencing has generally supported his claims on this front). He claims that there are no "intermediate forms" between the fantail and the pouter that possess a mix of the traits that make the two species unique; rather, there is a series of intermediate forms diverging from the rock-pigeon to the fantail and a separate series of intermediate forms diverging from the rock-pigeon to the pouter.
It may appear that Darwin is especially invested in pigeon taxonomy, but this is not necessarily the case. He uses the pigeons as a motif because they are an easily-explained and easily-understood case study for many of the broader ideas Darwin is trying to put forth about evolution and natural selection. The rock pigeon may predate the fantail and the pouter, but rock pigeons (or rock doves, as they are more often called today) are still around. In fact, they are ubiquitous. Darwin's readers could easily call up the image of all three varieties of birds and understand what he was saying about the way one species morphed into three over time. By returning often to this same clear example, Darwin offers his readers a touchstone in a complicated argument. Each time pigeons come up, the reader can return to a point of understanding from which Darwin can then build further.